Hamiltonian graph

A useful condition both necessary and sufficient for a graph to be Hamiltonian is not known. Ore’s theorem and the Bondy and Chvátal theorem give sufficient conditions, while a necessary condition follows quickly from the definition, namely:

Let G=(V,E) be a graph of order at least 3. If G is Hamiltonian, then for every proper subsetU of V, the subgraph induced by V-U has at most |U|components.